We give an asymptotic probabilistic real Riemann–Hurwitz formula computing the expected real ramification index of a random covering over the Riemann sphere. More generally, we study the asymptotic expected number and distribution of critical points of a random real Lefschetz pencil over a smooth real algebraic variety. Throughout the paper, we give similar results for the complex case. Our main tool is Hörmander theory of peak sections.
Dans cet article, on donne une formule de Riemann–Hurwitz asymptotique et probabiliste qui calcule la valeur attendue de l’indice de ramification réel d’un revêtement aléatoire de la sphère de Riemann. Plus généralement, on étudie l’asymptotique de la valeur attendue du nombre et de la distribution des points critiques réels d’un pinceau de Lefschetz réel sur une variété algébrique réelle. Tout au long de l’article, on donne des résultats analogues pour le cas complexe. Notre outil principal est la théorie des sections pics d’Hörmander.
Revised:
Accepted:
Published online:
Keywords: real algebraic varieties, Lefschetz pencils, peak sections, random geometry
Mot clés : variétés algébriques réelles, pinceaux de Lefschetz, sections pics, géométrie aléatoire
Ancona, Michele 1
@article{AIF_2020__70_3_1085_0, author = {Ancona, Michele}, title = {Expected number and distribution of critical points of real {Lefschetz} pencils}, journal = {Annales de l'Institut Fourier}, pages = {1085--1113}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {3}, year = {2020}, doi = {10.5802/aif.3331}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3331/} }
TY - JOUR AU - Ancona, Michele TI - Expected number and distribution of critical points of real Lefschetz pencils JO - Annales de l'Institut Fourier PY - 2020 SP - 1085 EP - 1113 VL - 70 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3331/ DO - 10.5802/aif.3331 LA - en ID - AIF_2020__70_3_1085_0 ER -
%0 Journal Article %A Ancona, Michele %T Expected number and distribution of critical points of real Lefschetz pencils %J Annales de l'Institut Fourier %D 2020 %P 1085-1113 %V 70 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3331/ %R 10.5802/aif.3331 %G en %F AIF_2020__70_3_1085_0
Ancona, Michele. Expected number and distribution of critical points of real Lefschetz pencils. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1085-1113. doi : 10.5802/aif.3331. https://aif.centre-mersenne.org/articles/10.5802/aif.3331/
[1] Average Euler characteristic of random real algebraic varieties, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 9, pp. 507-512 | DOI | MR
[2] Geometric measure theory, Grundlehren der Mathematischen Wissenschaften, 153, Springer, 1969, xiv+676 pages | MR
[3] What is the total Betti number of a random real hypersurface?, J. Reine Angew. Math., Volume 689 (2014), pp. 137-168 | DOI | MR
[4] Expected topology of random real algebraic submanifolds, J. Inst. Math. Jussieu, Volume 14 (2015) no. 4, pp. 673-702 | DOI | MR
[5] Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, J. Eur. Math. Soc., Volume 18 (2016) no. 4, pp. 733-772 | DOI | MR
[6] An introduction to complex analysis in several variables, North-Holland Mathematical Library, 7, North-Holland, 1990, xii+254 pages | MR
[7] A correction to “On the average number of real roots of a random algebraic equation”, Bull. Am. Math. Soc., Volume 49 (1943), p. 938 | DOI | MR
[8] On the distribution of roots of random polynomials, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, 1993, pp. 419-431 | MR
[9] Statistics on Hilbert’s 16th problem, Int. Math. Res. Not. (2015) no. 12, pp. 4293-4321 | DOI | MR
[10] Gap probabilities and Betti numbers of a random intersection of quadrics, Discrete Comput. Geom., Volume 55 (2016) no. 2, pp. 462-496
[11] Expected volume and Euler characteristic of random submanifolds, J. Funct. Anal., Volume 270 (2016) no. 8, pp. 3047-3110 | DOI | MR
[12] The arc length and topology of a random lemniscate, J. Lond. Math. Soc., II. Ser., Volume 96 (2017) no. 3, pp. 621-641 | Zbl
[13] Critical sets of random smooth functions on compact manifolds, Asian J. Math., Volume 19 (2015) no. 3, pp. 391-432 | DOI | MR
[14] On the Euler characteristic of a random algebraic hypersurface, Zap. Nauchn. Semin. (POMI), Volume 252 (1998), pp. 224-230 | DOI | MR | Zbl
[15] Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys., Volume 200 (1999) no. 3, pp. 661-683 | DOI | MR
[16] Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992) (Progress in Mathematics), Volume 109, Birkhäuser, 1993, pp. 267-285 | DOI | MR | Zbl
[17] On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom., Volume 32 (1990) no. 1, pp. 99-130 | MR
Cited by Sources: